I don't know how low your version of low carb is, but... Ive been getting progressively lower for about 9 months now. Started at 60-80g, then reduced to 40-60g, then less than 40, and now 20ish. Felt better with every step.
So, the reason for that preamble, was
please, don't be afraid of the fat.
(although, only do more fat if you are v low carbing. fat and carbs REALLY don't make a good combo. that's when the weight gain and the triclycerides, and the cholesterol go wangy)
All my opinion and experience, but not for everyone, I do realise.